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Take position “1” where the first sample is taken. The voltage is at 0V. The
quantisation level is 0V which is encoded to 0100. The quantised signal after the first
sample is made to hold at 0V until the next sample is taken. The next sample is taken
at point “2” on the graph. When the next sample is taken at point “2” the analogue
voltage is approximately 0.5V. The nearest quantisation level is 0.5V. The analogue
signal at “2” snaps-and-holds to 0.5V. Which is mapped/encoded to 0110. The 0.5V
holds until the next sample is taken at “3”. At “3” the analogue signal has reached
1V. The nearest quantised level is 1V. There is now another snap-and-hold at 1V.
This level is mapped to binary 1000.

At sample 4 the analogue signal is above 1V. However, the nearest quantised level
is still 1V. Again, at sample “5” the analogue voltage is just slightly above 1V, but the
quantised signal is pulled to 1V. The next sample is taken at “6” and the quantised
signal snaps-and-holds to 0.75 voltage, which is the nearest permitted quantisation
level. The next sample is taken at “8”. The analogue voltage is 0V, and the nearest
quantised level is 0V. It should be understood that the analogue signal will always
be forced to go to the nearest quantisation level.

Sampling converted the continuous in time analogue signal to a digital signal that is
discrete in time. Quantisation has now converted the continuous in amplitude signal
to a digital signal discrete in time. Sampling and quantisation are performed by an
analogue to digital converter (ADC). After the ADC we now have a discrete digital
signal.

After digitisation, we can, if needed, perform mathematical operations on the signal
as it is now in binary format. A digital to analogue converter (DAC) can convert our
signal back to analogue.

In the example of figure 35-21, we had eight levels of quantisation. If we wanted
more accuracy or resolution, we would have to use a greater number of levels.

FOURIER TRANSFORM

The Fourier transform is an algorithm that converts signals in the time domain to the
frequency domain. A good example of this is the waterfall display (bandscope) on
some radio receivers. The vertical (y) axis of the waterfall is amplitude, and the
horizontal (x) axis is frequency. Oscilloscopes display signals in the “time” domain.
Some modern oscilloscopes have a built in Fast Fourier Transform (FFT) that allows
the display of signals in the “frequency” domain.

Gauss (Johann Carl Friedrich) was the first to propose the technique for calculating
the coefficients in a trigonometric of an asteroid’s orbit in 1805. However, it was not
until 1965 that a seminal paper by Cooley and Tukey caught the attention of the
science and engineering community, which also laid the foundation for the
discipline of digital signal processing.
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